ab-angle->ABCF B

Percentage Accurate: 53.9% → 63.5%

Time: 8.0s

Alternatives: 22

Speedup: 4.0×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 63.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_2 := \cos t\_1\\ t_3 := \sin t\_1\\ t_4 := t\_3 \cdot t\_2\\ t_5 := -t\_1\\ \mathbf{if}\;b\_m \leq 4.8 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 1\right)\\ \mathbf{elif}\;b\_m \leq 9.5 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2 \cdot t\_2, t\_3 \cdot 0, \left(-2 \cdot a\right) \cdot t\_4\right), a, \left(\left(b\_m \cdot b\_m\right) \cdot 2\right) \cdot t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b\_m, \frac{\sin \left(t\_1 - t\_5\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_5\right)\right)}{2}, 0\right), b\_m, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (* (* 0.005555555555555556 angle) PI))
        (t_2 (cos t_1))
        (t_3 (sin t_1))
        (t_4 (* t_3 t_2))
        (t_5 (- t_1)))
   (if (<= b_m 4.8e-60)
     (* (* (* (+ b_m a) (- b_m a)) 2.0) (* (sin (* PI (/ angle 180.0))) 1.0))
     (if (<= b_m 9.5e+137)
       (fma
        (fma (* 2.0 t_2) (* t_3 0.0) (* (* -2.0 a) t_4))
        a
        (* (* (* b_m b_m) 2.0) t_4))
       (fma
        (*
         2.0
         (fma
          b_m
          (/
           (+
            (sin (- t_1 t_5))
            (sin (fma (* 0.005555555555555556 angle) PI t_5)))
           2.0)
          0.0))
        b_m
        (* (* -2.0 (* a a)) (* (sin t_0) (cos t_0))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_2 = cos(t_1);
	double t_3 = sin(t_1);
	double t_4 = t_3 * t_2;
	double t_5 = -t_1;
	double tmp;
	if (b_m <= 4.8e-60) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (sin((((double) M_PI) * (angle / 180.0))) * 1.0);
	} else if (b_m <= 9.5e+137) {
		tmp = fma(fma((2.0 * t_2), (t_3 * 0.0), ((-2.0 * a) * t_4)), a, (((b_m * b_m) * 2.0) * t_4));
	} else {
		tmp = fma((2.0 * fma(b_m, ((sin((t_1 - t_5)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_5))) / 2.0), 0.0)), b_m, ((-2.0 * (a * a)) * (sin(t_0) * cos(t_0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_2 = cos(t_1)
	t_3 = sin(t_1)
	t_4 = Float64(t_3 * t_2)
	t_5 = Float64(-t_1)
	tmp = 0.0
	if (b_m <= 4.8e-60)
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(sin(Float64(pi * Float64(angle / 180.0))) * 1.0));
	elseif (b_m <= 9.5e+137)
		tmp = fma(fma(Float64(2.0 * t_2), Float64(t_3 * 0.0), Float64(Float64(-2.0 * a) * t_4)), a, Float64(Float64(Float64(b_m * b_m) * 2.0) * t_4));
	else
		tmp = fma(Float64(2.0 * fma(b_m, Float64(Float64(sin(Float64(t_1 - t_5)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_5))) / 2.0), 0.0)), b_m, Float64(Float64(-2.0 * Float64(a * a)) * Float64(sin(t_0) * cos(t_0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$2), $MachinePrecision]}, Block[{t$95$5 = (-t$95$1)}, If[LessEqual[b$95$m, 4.8e-60], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 9.5e+137], N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(t$95$3 * 0.0), $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b$95$m * N[(N[(N[Sin[N[(t$95$1 - t$95$5), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] * b$95$m + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.80000000000000019e-60

   1. Initial program 62.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 60.0%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right) \]

   if 4.80000000000000019e-60 < b < 9.50000000000000031e137

   1. Initial program 53.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{2 \cdot \left({b}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + a \cdot \left(-2 \cdot \left(a \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]

   9. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2 \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 0, \left(-2 \cdot a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right), a, \left(\left(b \cdot b\right) \cdot 2\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)} \]

   if 9.50000000000000031e137 < b

   1. Initial program 41.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      2. lift-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      5. sin-cos-mult N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

   8. Applied rewrites 65.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi - \left(-\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi - \left(-\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right)}{2}, 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 65.0%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi - \left(-\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}, 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\\ t_1 := \pi \cdot \frac{angle}{180}\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(angle \cdot \pi\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \cos t\_1\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\_m\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right), 2, \left(\pi \cdot b\_m\right) \cdot 0.011111111111111112\right) \cdot angle, b\_m, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin t\_2 \cdot \cos t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\sin t\_1 \cdot \sin \left(\left(-\frac{angle}{180} \cdot \pi\right) + \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* 2.0 (- (pow b_m 2.0) (pow a 2.0))))
        (t_1 (* PI (/ angle 180.0)))
        (t_2 (* (* PI angle) 0.005555555555555556)))
   (if (<= t_0 -5e+302)
     (fma
      (* (* -0.011111111111111112 a) (* angle PI))
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (if (<= t_0 2e+290)
       (*
        (* (* (+ b_m a) (- b_m a)) 2.0)
        (* (sin (/ (* PI angle) 180.0)) (cos t_1)))
       (if (<= t_0 INFINITY)
         (fma
          (*
           (fma
            (*
             (* (* angle angle) b_m)
             (* (* (* PI PI) PI) -1.1431184270690443e-7))
            2.0
            (* (* PI b_m) 0.011111111111111112))
           angle)
          b_m
          (* (* -2.0 (* a a)) (* (sin t_2) (cos t_2))))
         (*
          (* (* a (- b_m a)) 2.0)
          (* (sin t_1) (sin (+ (- (* (/ angle 180.0) PI)) (/ PI 2.0))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = 2.0 * (pow(b_m, 2.0) - pow(a, 2.0));
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (t_0 <= -5e+302) {
		tmp = fma(((-0.011111111111111112 * a) * (angle * ((double) M_PI))), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else if (t_0 <= 2e+290) {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (sin(((((double) M_PI) * angle) / 180.0)) * cos(t_1));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((fma((((angle * angle) * b_m) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * -1.1431184270690443e-7)), 2.0, ((((double) M_PI) * b_m) * 0.011111111111111112)) * angle), b_m, ((-2.0 * (a * a)) * (sin(t_2) * cos(t_2))));
	} else {
		tmp = ((a * (b_m - a)) * 2.0) * (sin(t_1) * sin((-((angle / 180.0) * ((double) M_PI)) + (((double) M_PI) / 2.0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(pi * Float64(angle / 180.0))
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (t_0 <= -5e+302)
		tmp = fma(Float64(Float64(-0.011111111111111112 * a) * Float64(angle * pi)), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	elseif (t_0 <= 2e+290)
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(sin(Float64(Float64(pi * angle) / 180.0)) * cos(t_1)));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(fma(Float64(Float64(Float64(angle * angle) * b_m) * Float64(Float64(Float64(pi * pi) * pi) * -1.1431184270690443e-7)), 2.0, Float64(Float64(pi * b_m) * 0.011111111111111112)) * angle), b_m, Float64(Float64(-2.0 * Float64(a * a)) * Float64(sin(t_2) * cos(t_2))));
	else
		tmp = Float64(Float64(Float64(a * Float64(b_m - a)) * 2.0) * Float64(sin(t_1) * sin(Float64(Float64(-Float64(Float64(angle / 180.0) * pi)) + Float64(pi / 2.0)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+302], N[(N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+290], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(N[(N[(angle * angle), $MachinePrecision] * b$95$m), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(Pi * b$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * b$95$m + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$2], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[N[((-N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -5e302

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 49.8

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 69.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      5. lift-PI.f64 69.6

         \[\leadsto \mathsf{fma}\left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(angle \cdot \pi\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   10. Applied rewrites 69.6%

       \[\leadsto \mathsf{fma}\left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(angle \cdot \pi\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if -5e302 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 2.00000000000000012e290

   1. Initial program 59.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \mathsf{PI}\left(\right)}}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \mathsf{PI}\left(\right)}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot angle}}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot angle}}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-PI.f64 59.8

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi} \cdot angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. Applied rewrites 59.8%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   if 2.00000000000000012e290 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 54.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(angle \cdot \left(\frac{1}{90} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(b \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{90} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(b \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \cdot angle, b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{90} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(b \cdot \left(\frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \cdot angle, b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

   9. Applied rewrites 72.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(angle \cdot angle\right) \cdot b\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right), 2, \left(\pi \cdot b\right) \cdot 0.011111111111111112\right) \cdot angle, b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   if +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 0.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in a around inf

      \[\leadsto \left(\left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 49.7%

      \[\leadsto \left(\left(\color{blue}{a} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{angle}{180}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\color{blue}{\frac{angle}{180} \cdot \mathsf{PI}\left(\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\color{blue}{\frac{angle}{180} \cdot \mathsf{PI}\left(\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\frac{angle}{180} \cdot \color{blue}{\pi}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\frac{angle}{180} \cdot \pi\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      13. lift-PI.f64 48.5

          \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\frac{angle}{180} \cdot \pi\right) + \frac{\color{blue}{\pi}}{2}\right)\right) \]

   8. Applied rewrites 48.5%

      \[\leadsto \left(\left(a \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(-\frac{angle}{180} \cdot \pi\right) + \frac{\pi}{2}\right)}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 62.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := -t\_0\\ t_2 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b\_m, \frac{\sin \left(t\_0 - t\_1\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, t\_1\right)\right)}{2}, 0\right), b\_m, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin t\_2 \cdot \cos t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (- t_0))
        (t_2 (* (* PI angle) 0.005555555555555556)))
   (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -4e+114)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (fma
      (*
       2.0
       (fma
        b_m
        (/
         (+
          (sin (- t_0 t_1))
          (sin (fma (* 0.005555555555555556 angle) PI t_1)))
         2.0)
        0.0))
      b_m
      (* (* -2.0 (* a a)) (* (sin t_2) (cos t_2)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = -t_0;
	double t_2 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -4e+114) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = fma((2.0 * fma(b_m, ((sin((t_0 - t_1)) + sin(fma((0.005555555555555556 * angle), ((double) M_PI), t_1))) / 2.0), 0.0)), b_m, ((-2.0 * (a * a)) * (sin(t_2) * cos(t_2))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -4e+114)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = fma(Float64(2.0 * fma(b_m, Float64(Float64(sin(Float64(t_0 - t_1)) + sin(fma(Float64(0.005555555555555556 * angle), pi, t_1))) / 2.0), 0.0)), b_m, Float64(Float64(-2.0 * Float64(a * a)) * Float64(sin(t_2) * cos(t_2))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+114], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b$95$m * N[(N[(N[Sin[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] * b$95$m + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$2], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4e114

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 49.2

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 62.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if -4e114 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      2. lift-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      5. sin-cos-mult N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

   8. Applied rewrites 63.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi - \left(-\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi - \left(-\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(\frac{1}{180} \cdot angle, \pi, -\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right)}{2}, 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 63.0%

       \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi - \left(-\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}, 0\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b\_m}^{2} - {a}^{2}\right)\right) \cdot \sin t\_2\right) \cdot \cos t\_2 \leq 10^{-172}:\\ \;\;\;\;\left(2 \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-1, a \cdot t\_0, t\_0 \cdot \left(b\_m - 1 \cdot b\_m\right)\right), \left(b\_m \cdot b\_m\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), b\_m, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin t\_1 \cdot \cos t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) (sin t_2)) (cos t_2))
        1e-172)
     (*
      (* 2.0 (cos (* (* 0.005555555555555556 angle) PI)))
      (fma
       a
       (fma -1.0 (* a t_0) (* t_0 (- b_m (* 1.0 b_m))))
       (* (* b_m b_m) t_0)))
     (fma
      (* b_m (sin (* 0.011111111111111112 (* angle PI))))
      b_m
      (* (* -2.0 (* a a)) (* (sin t_1) (cos t_1)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) * sin(t_2)) * cos(t_2)) <= 1e-172) {
		tmp = (2.0 * cos(((0.005555555555555556 * angle) * ((double) M_PI)))) * fma(a, fma(-1.0, (a * t_0), (t_0 * (b_m - (1.0 * b_m)))), ((b_m * b_m) * t_0));
	} else {
		tmp = fma((b_m * sin((0.011111111111111112 * (angle * ((double) M_PI))))), b_m, ((-2.0 * (a * a)) * (sin(t_1) * cos(t_1))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) * sin(t_2)) * cos(t_2)) <= 1e-172)
		tmp = Float64(Float64(2.0 * cos(Float64(Float64(0.005555555555555556 * angle) * pi))) * fma(a, fma(-1.0, Float64(a * t_0), Float64(t_0 * Float64(b_m - Float64(1.0 * b_m)))), Float64(Float64(b_m * b_m) * t_0)));
	else
		tmp = fma(Float64(b_m * sin(Float64(0.011111111111111112 * Float64(angle * pi)))), b_m, Float64(Float64(-2.0 * Float64(a * a)) * Float64(sin(t_1) * cos(t_1))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision], 1e-172], N[(N[(2.0 * N[Cos[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * N[(-1.0 * N[(a * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(b$95$m - N[(1.0 * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 1e-172

   1. Initial program 61.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 64.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   7. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   9. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \left(2 \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)\right) + \color{blue}{{b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \left(2 \cdot \cos \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)\right) \cdot \mathsf{fma}\left(a, -1 \cdot \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(b + -1 \cdot b\right)}, {b}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   12. Applied rewrites 65.4%

       \[\leadsto \left(2 \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(-1, a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b - 1 \cdot b\right)\right)}, \left(b \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   if 1e-172 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 44.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      2. lift-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      5. sin-cos-mult N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

   8. Applied rewrites 54.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi - \left(-\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. lift-PI.f64 54.1

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   11. Applied rewrites 54.1%

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq -4 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), b\_m, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin t\_0 \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) -4e+114)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (fma
      (* b_m (sin (* 0.011111111111111112 (* angle PI))))
      b_m
      (* (* -2.0 (* a a)) (* (sin t_0) (cos t_0)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= -4e+114) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = fma((b_m * sin((0.011111111111111112 * (angle * ((double) M_PI))))), b_m, ((-2.0 * (a * a)) * (sin(t_0) * cos(t_0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= -4e+114)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = fma(Float64(b_m * sin(Float64(0.011111111111111112 * Float64(angle * pi)))), b_m, Float64(Float64(-2.0 * Float64(a * a)) * Float64(sin(t_0) * cos(t_0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+114], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m * N[Sin[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -4e114

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 49.2

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 62.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if -4e114 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 59.3%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      2. lift-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      5. sin-cos-mult N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} - \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) + \sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180} + \left(\mathsf{neg}\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

   8. Applied rewrites 63.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi - \left(-\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) + \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, -\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)}{2}, \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)\right)\right) \]

       5. lift-PI.f64 63.1

          \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]

   11. Applied rewrites 63.1%

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;angle \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \pi \cdot angle, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b\_m, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\_m\right)\\ \mathbf{elif}\;angle \leq 4.3 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\sin t\_0 \cdot \sin \left(\left(-t\_0\right) + \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= angle 3e-191)
     (fma
      (* -0.011111111111111112 (* a a))
      (* PI angle)
      (*
       (*
        0.011111111111111112
        (fma (* PI b_m) angle (* (* (* 0.0 a) PI) angle)))
       b_m))
     (if (<= angle 4.3e+51)
       (fma
        (fma (* (* PI angle) a) -0.011111111111111112 0.0)
        a
        (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
       (*
        (* (* (+ b_m a) (- b_m a)) 2.0)
        (* (sin t_0) (sin (+ (- t_0) (/ PI 2.0)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if (angle <= 3e-191) {
		tmp = fma((-0.011111111111111112 * (a * a)), (((double) M_PI) * angle), ((0.011111111111111112 * fma((((double) M_PI) * b_m), angle, (((0.0 * a) * ((double) M_PI)) * angle))) * b_m));
	} else if (angle <= 4.3e+51) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (sin(t_0) * sin((-t_0 + (((double) M_PI) / 2.0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (angle <= 3e-191)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(pi * angle), Float64(Float64(0.011111111111111112 * fma(Float64(pi * b_m), angle, Float64(Float64(Float64(0.0 * a) * pi) * angle))) * b_m));
	elseif (angle <= 4.3e+51)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(sin(t_0) * sin(Float64(Float64(-t_0) + Float64(pi / 2.0)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 3e-191], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision] + N[(N[(0.011111111111111112 * N[(N[(Pi * b$95$m), $MachinePrecision] * angle + N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 4.3e+51], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[((-t$95$0) + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0000000000000001e-191

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 56.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + b \cdot \left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \color{blue}{\mathsf{PI}\left(\right)}, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\pi \cdot angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \]

   if 3.0000000000000001e-191 < angle < 4.2999999999999997e51

   1. Initial program 74.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 76.3

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 78.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 4.2999999999999997e51 < angle

   1. Initial program 29.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 32.1%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \frac{angle}{180}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\color{blue}{\left(-\pi \cdot \frac{angle}{180}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

      8. lift-PI.f64 31.4

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\color{blue}{\pi}}{2}\right)\right) \]

   5. Applied rewrites 31.4%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\sin \left(\left(-\pi \cdot \frac{angle}{180}\right) + \frac{\pi}{2}\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \pi \cdot angle, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b\_m, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\_m\right)\\ \mathbf{elif}\;angle \leq 1.7 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 3e-191)
   (fma
    (* -0.011111111111111112 (* a a))
    (* PI angle)
    (*
     (* 0.011111111111111112 (fma (* PI b_m) angle (* (* (* 0.0 a) PI) angle)))
     b_m))
   (if (<= angle 1.7e-42)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (*
      (* (* (+ b_m a) (- b_m a)) 2.0)
      (* (sin (* PI (/ angle 180.0))) 1.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 3e-191) {
		tmp = fma((-0.011111111111111112 * (a * a)), (((double) M_PI) * angle), ((0.011111111111111112 * fma((((double) M_PI) * b_m), angle, (((0.0 * a) * ((double) M_PI)) * angle))) * b_m));
	} else if (angle <= 1.7e-42) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = (((b_m + a) * (b_m - a)) * 2.0) * (sin((((double) M_PI) * (angle / 180.0))) * 1.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 3e-191)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(pi * angle), Float64(Float64(0.011111111111111112 * fma(Float64(pi * b_m), angle, Float64(Float64(Float64(0.0 * a) * pi) * angle))) * b_m));
	elseif (angle <= 1.7e-42)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(Float64(b_m + a) * Float64(b_m - a)) * 2.0) * Float64(sin(Float64(pi * Float64(angle / 180.0))) * 1.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 3e-191], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision] + N[(N[(0.011111111111111112 * N[(N[(Pi * b$95$m), $MachinePrecision] * angle + N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.7e-42], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0000000000000001e-191

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 56.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + b \cdot \left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \color{blue}{\mathsf{PI}\left(\right)}, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\pi \cdot angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \]

   if 3.0000000000000001e-191 < angle < 1.70000000000000011e-42

   1. Initial program 80.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 86.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 90.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 1.70000000000000011e-42 < angle

   1. Initial program 38.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 38.7%

      \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \pi \cdot angle, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b\_m, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\_m\right)\\ \mathbf{elif}\;angle \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 3e-191)
   (fma
    (* -0.011111111111111112 (* a a))
    (* PI angle)
    (*
     (* 0.011111111111111112 (fma (* PI b_m) angle (* (* (* 0.0 a) PI) angle)))
     b_m))
   (if (<= angle 2.15e+51)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (* (* (- b_m a) (+ a b_m)) (sin (* 2.0 (* PI (/ angle 180.0))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 3e-191) {
		tmp = fma((-0.011111111111111112 * (a * a)), (((double) M_PI) * angle), ((0.011111111111111112 * fma((((double) M_PI) * b_m), angle, (((0.0 * a) * ((double) M_PI)) * angle))) * b_m));
	} else if (angle <= 2.15e+51) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = ((b_m - a) * (a + b_m)) * sin((2.0 * (((double) M_PI) * (angle / 180.0))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 3e-191)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(pi * angle), Float64(Float64(0.011111111111111112 * fma(Float64(pi * b_m), angle, Float64(Float64(Float64(0.0 * a) * pi) * angle))) * b_m));
	elseif (angle <= 2.15e+51)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(b_m - a) * Float64(a + b_m)) * sin(Float64(2.0 * Float64(pi * Float64(angle / 180.0)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 3e-191], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision] + N[(N[(0.011111111111111112 * N[(N[(Pi * b$95$m), $MachinePrecision] * angle + N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 2.15e+51], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0000000000000001e-191

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 56.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + b \cdot \left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \color{blue}{\mathsf{PI}\left(\right)}, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\pi \cdot angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \]

   if 3.0000000000000001e-191 < angle < 2.1499999999999999e51

   1. Initial program 74.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 76.3

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 78.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 2.1499999999999999e51 < angle

   1. Initial program 29.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 32.1%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right)} \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)}\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \]

   5. Applied rewrites 32.1%

      \[\leadsto \color{blue}{\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \pi \cdot angle, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b\_m, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\_m\right)\\ \mathbf{elif}\;angle \leq 1.7 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 3e-191)
   (fma
    (* -0.011111111111111112 (* a a))
    (* PI angle)
    (*
     (* 0.011111111111111112 (fma (* PI b_m) angle (* (* (* 0.0 a) PI) angle)))
     b_m))
   (if (<= angle 1.7e-42)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (*
      2.0
      (*
       (sin (* (* 0.005555555555555556 angle) PI))
       (* (- b_m a) (+ a b_m)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 3e-191) {
		tmp = fma((-0.011111111111111112 * (a * a)), (((double) M_PI) * angle), ((0.011111111111111112 * fma((((double) M_PI) * b_m), angle, (((0.0 * a) * ((double) M_PI)) * angle))) * b_m));
	} else if (angle <= 1.7e-42) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = 2.0 * (sin(((0.005555555555555556 * angle) * ((double) M_PI))) * ((b_m - a) * (a + b_m)));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 3e-191)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(pi * angle), Float64(Float64(0.011111111111111112 * fma(Float64(pi * b_m), angle, Float64(Float64(Float64(0.0 * a) * pi) * angle))) * b_m));
	elseif (angle <= 1.7e-42)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(2.0 * Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * Float64(Float64(b_m - a) * Float64(a + b_m))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 3e-191], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision] + N[(N[(0.011111111111111112 * N[(N[(Pi * b$95$m), $MachinePrecision] * angle + N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.7e-42], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0000000000000001e-191

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 56.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + b \cdot \left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \color{blue}{\mathsf{PI}\left(\right)}, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\pi \cdot angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \]

   if 3.0000000000000001e-191 < angle < 1.70000000000000011e-42

   1. Initial program 80.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 86.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 90.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 1.70000000000000011e-42 < angle

   1. Initial program 38.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

   6. Applied rewrites 41.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

   7. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   9. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 38.9%

       \[\leadsto 2 \cdot \left(\color{blue}{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)} \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \pi \cdot angle, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b\_m, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\_m\right)\\ \mathbf{elif}\;angle \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-2.2862368541380886 \cdot 10^{-7} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 3e-191)
   (fma
    (* -0.011111111111111112 (* a a))
    (* PI angle)
    (*
     (* 0.011111111111111112 (fma (* PI b_m) angle (* (* (* 0.0 a) PI) angle)))
     b_m))
   (if (<= angle 8e+109)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (*
      -2.2862368541380886e-7
      (*
       (* (* angle angle) angle)
       (* (* (* PI PI) PI) (* (+ a b_m) (- b_m a))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 3e-191) {
		tmp = fma((-0.011111111111111112 * (a * a)), (((double) M_PI) * angle), ((0.011111111111111112 * fma((((double) M_PI) * b_m), angle, (((0.0 * a) * ((double) M_PI)) * angle))) * b_m));
	} else if (angle <= 8e+109) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = -2.2862368541380886e-7 * (((angle * angle) * angle) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * ((a + b_m) * (b_m - a))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 3e-191)
		tmp = fma(Float64(-0.011111111111111112 * Float64(a * a)), Float64(pi * angle), Float64(Float64(0.011111111111111112 * fma(Float64(pi * b_m), angle, Float64(Float64(Float64(0.0 * a) * pi) * angle))) * b_m));
	elseif (angle <= 8e+109)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(-2.2862368541380886e-7 * Float64(Float64(Float64(angle * angle) * angle) * Float64(Float64(Float64(pi * pi) * pi) * Float64(Float64(a + b_m) * Float64(b_m - a)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 3e-191], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision] + N[(N[(0.011111111111111112 * N[(N[(Pi * b$95$m), $MachinePrecision] * angle + N[(N[(N[(0.0 * a), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 8e+109], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(-2.2862368541380886e-7 * N[(N[(N[(angle * angle), $MachinePrecision] * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0000000000000001e-191

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 56.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + b \cdot \left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \color{blue}{\mathsf{PI}\left(\right)}, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot {a}^{2}, angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), angle \cdot \mathsf{PI}\left(\right), b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \mathsf{PI}\left(\right) \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot a\right), \pi \cdot angle, \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) \cdot b\right) \]

   7. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(-0.011111111111111112 \cdot \left(a \cdot a\right), \color{blue}{\pi \cdot angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, \left(\left(0 \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \]

   if 3.0000000000000001e-191 < angle < 7.99999999999999985e109

   1. Initial program 65.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 66.6

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 67.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 7.99999999999999985e109 < angle

   1. Initial program 29.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

   6. Applied rewrites 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \pi, \left(b - a\right) \cdot \left(a + b\right), \left(\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot 2\right) \cdot angle} \]

   7. Taylor expanded in angle around inf

      \[\leadsto \frac{-1}{4374000} \cdot \color{blue}{\left({angle}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left({angle}^{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left({angle}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. unpow3 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left({angle}^{2} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left({angle}^{2} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      9. pow3 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied rewrites 22.7%

      \[\leadsto -2.2862368541380886 \cdot 10^{-7} \cdot \color{blue}{\left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot -0.011111111111111112, angle, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b\_m, angle, 0 \cdot angle\right)\right) \cdot b\_m\right)\\ \mathbf{elif}\;angle \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-2.2862368541380886 \cdot 10^{-7} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 3e-191)
   (fma
    (* (* (* a a) PI) -0.011111111111111112)
    angle
    (* (* 0.011111111111111112 (fma (* PI b_m) angle (* 0.0 angle))) b_m))
   (if (<= angle 8e+109)
     (fma
      (fma (* (* PI angle) a) -0.011111111111111112 0.0)
      a
      (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
     (*
      -2.2862368541380886e-7
      (*
       (* (* angle angle) angle)
       (* (* (* PI PI) PI) (* (+ a b_m) (- b_m a))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 3e-191) {
		tmp = fma((((a * a) * ((double) M_PI)) * -0.011111111111111112), angle, ((0.011111111111111112 * fma((((double) M_PI) * b_m), angle, (0.0 * angle))) * b_m));
	} else if (angle <= 8e+109) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = -2.2862368541380886e-7 * (((angle * angle) * angle) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * ((a + b_m) * (b_m - a))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 3e-191)
		tmp = fma(Float64(Float64(Float64(a * a) * pi) * -0.011111111111111112), angle, Float64(Float64(0.011111111111111112 * fma(Float64(pi * b_m), angle, Float64(0.0 * angle))) * b_m));
	elseif (angle <= 8e+109)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(-2.2862368541380886e-7 * Float64(Float64(Float64(angle * angle) * angle) * Float64(Float64(Float64(pi * pi) * pi) * Float64(Float64(a + b_m) * Float64(b_m - a)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 3e-191], N[(N[(N[(N[(a * a), $MachinePrecision] * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * angle + N[(N[(0.011111111111111112 * N[(N[(Pi * b$95$m), $MachinePrecision] * angle + N[(0.0 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 8e+109], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(-2.2862368541380886e-7 * N[(N[(N[(angle * angle), $MachinePrecision] * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.0000000000000001e-191

   1. Initial program 55.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 56.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 37.2

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 37.2%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{b \cdot \left(\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]

   10. Applied rewrites 57.2%

       \[\leadsto \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot -0.011111111111111112, \color{blue}{angle}, \left(0.011111111111111112 \cdot \mathsf{fma}\left(\pi \cdot b, angle, 0 \cdot angle\right)\right) \cdot b\right) \]

   if 3.0000000000000001e-191 < angle < 7.99999999999999985e109

   1. Initial program 65.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 66.6

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 67.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 7.99999999999999985e109 < angle

   1. Initial program 29.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

   6. Applied rewrites 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \pi, \left(b - a\right) \cdot \left(a + b\right), \left(\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot 2\right) \cdot angle} \]

   7. Taylor expanded in angle around inf

      \[\leadsto \frac{-1}{4374000} \cdot \color{blue}{\left({angle}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left({angle}^{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left({angle}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. unpow3 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left({angle}^{2} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left({angle}^{2} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      9. pow3 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied rewrites 22.7%

      \[\leadsto -2.2862368541380886 \cdot 10^{-7} \cdot \color{blue}{\left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;-2.2862368541380886 \cdot 10^{-7} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 8e+109)
   (fma
    (fma (* (* PI angle) a) -0.011111111111111112 0.0)
    a
    (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
   (*
    -2.2862368541380886e-7
    (*
     (* (* angle angle) angle)
     (* (* (* PI PI) PI) (* (+ a b_m) (- b_m a)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 8e+109) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = -2.2862368541380886e-7 * (((angle * angle) * angle) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * ((a + b_m) * (b_m - a))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 8e+109)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(-2.2862368541380886e-7 * Float64(Float64(Float64(angle * angle) * angle) * Float64(Float64(Float64(pi * pi) * pi) * Float64(Float64(a + b_m) * Float64(b_m - a)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 8e+109], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(-2.2862368541380886e-7 * N[(N[(N[(angle * angle), $MachinePrecision] * angle), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.99999999999999985e109

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 59.2

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 7.99999999999999985e109 < angle

   1. Initial program 29.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

   6. Applied rewrites 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \pi, \left(b - a\right) \cdot \left(a + b\right), \left(\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot 2\right) \cdot angle} \]

   7. Taylor expanded in angle around inf

      \[\leadsto \frac{-1}{4374000} \cdot \color{blue}{\left({angle}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left({angle}^{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left({angle}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. unpow3 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left({angle}^{2} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left({angle}^{2} \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(\color{blue}{a} + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      9. pow3 N/A

         \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      12. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4374000} \cdot \left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - \color{blue}{a}\right)\right)\right)\right) \]

   9. Applied rewrites 22.7%

      \[\leadsto -2.2862368541380886 \cdot 10^{-7} \cdot \color{blue}{\left(\left(\left(angle \cdot angle\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 55.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 9.2 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(-0.011111111111111112, \pi, 2.2862368541380886 \cdot 10^{-7} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right) \cdot angle\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b\_m \cdot \left(b\_m - a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 8e+109)
   (fma
    (fma (* (* PI angle) a) -0.011111111111111112 0.0)
    a
    (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
   (if (<= angle 9.2e+195)
     (*
      (*
       (* a a)
       (fma
        -0.011111111111111112
        PI
        (* 2.2862368541380886e-7 (* (* angle angle) (* (* PI PI) PI)))))
      angle)
     (* (* 0.011111111111111112 angle) (* PI (* b_m (- b_m a)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 8e+109) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else if (angle <= 9.2e+195) {
		tmp = ((a * a) * fma(-0.011111111111111112, ((double) M_PI), (2.2862368541380886e-7 * ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))))) * angle;
	} else {
		tmp = (0.011111111111111112 * angle) * (((double) M_PI) * (b_m * (b_m - a)));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 8e+109)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	elseif (angle <= 9.2e+195)
		tmp = Float64(Float64(Float64(a * a) * fma(-0.011111111111111112, pi, Float64(2.2862368541380886e-7 * Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi))))) * angle);
	else
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(b_m * Float64(b_m - a))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 8e+109], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 9.2e+195], N[(N[(N[(a * a), $MachinePrecision] * N[(-0.011111111111111112 * Pi + N[(2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(b$95$m * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 7.99999999999999985e109

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 59.2

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 7.99999999999999985e109 < angle < 9.2000000000000005e195

   1. Initial program 29.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

   6. Applied rewrites 4.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \pi, \left(b - a\right) \cdot \left(a + b\right), \left(\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot 2\right) \cdot angle} \]

   7. Taylor expanded in a around inf

      \[\leadsto \left({a}^{2} \cdot \left(\frac{-1}{90} \cdot \mathsf{PI}\left(\right) + \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left({a}^{2} \cdot \left(\frac{-1}{90} \cdot \mathsf{PI}\left(\right) + \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      2. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{90} \cdot \mathsf{PI}\left(\right) + \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\frac{-1}{90} \cdot \mathsf{PI}\left(\right) + \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \mathsf{PI}\left(\right), \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      8. pow2 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \cdot angle \]

      10. pow3 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot angle \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot angle \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot angle \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot angle \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{-1}{90}, \pi, \frac{1}{4374000} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot angle \]

      15. lift-PI.f64 23.0

          \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(-0.011111111111111112, \pi, 2.2862368541380886 \cdot 10^{-7} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right) \cdot angle \]

   9. Applied rewrites 23.0%

      \[\leadsto \left(\left(a \cdot a\right) \cdot \mathsf{fma}\left(-0.011111111111111112, \pi, 2.2862368541380886 \cdot 10^{-7} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right) \cdot angle \]

   if 9.2000000000000005e195 < angle

   1. Initial program 29.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 26.1

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot \left(\color{blue}{b} - a\right)\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 27.0%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot \left(\color{blue}{b} - a\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.12 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right) \cdot angle\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= b_m 1.12e+165)
   (fma
    (fma (* (* PI angle) a) -0.011111111111111112 0.0)
    a
    (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
   (*
    (*
     (* b_m b_m)
     (fma
      -2.2862368541380886e-7
      (* (* angle angle) (* (* PI PI) PI))
      (* 0.011111111111111112 PI)))
    angle)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (b_m <= 1.12e+165) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = ((b_m * b_m) * fma(-2.2862368541380886e-7, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))), (0.011111111111111112 * ((double) M_PI)))) * angle;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (b_m <= 1.12e+165)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * fma(-2.2862368541380886e-7, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * pi)), Float64(0.011111111111111112 * pi))) * angle);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[b$95$m, 1.12e+165], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(-2.2862368541380886e-7 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] + N[(0.011111111111111112 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1200000000000001e165

   1. Initial program 57.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 52.9

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 56.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 56.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 1.1200000000000001e165 < b

   1. Initial program 41.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      8. lift-PI.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      14. lift-/.f64 N/A

          \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

   3. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\right) \cdot \color{blue}{angle} \]

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.011111111111111112 \cdot \pi, \left(b - a\right) \cdot \left(a + b\right), \left(\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right) \cdot \left(angle \cdot angle\right)\right) \cdot 2\right) \cdot angle} \]

   7. Taylor expanded in b around inf

      \[\leadsto \left({b}^{2} \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left({b}^{2} \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      2. unpow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, {angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      6. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}, \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      8. pow3 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle \]

      15. lift-PI.f64 51.3

          \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right) \cdot angle \]

   9. Applied rewrites 51.3%

      \[\leadsto \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(-2.2862368541380886 \cdot 10^{-7}, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 0.011111111111111112 \cdot \pi\right)\right) \cdot angle \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 54.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b\_m + a\right) \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 8e+109)
   (fma
    (fma (* (* PI angle) a) -0.011111111111111112 0.0)
    a
    (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
   (* (* 0.011111111111111112 angle) (* PI (* (+ b_m a) b_m)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 8e+109) {
		tmp = fma(fma(((((double) M_PI) * angle) * a), -0.011111111111111112, 0.0), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = (0.011111111111111112 * angle) * (((double) M_PI) * ((b_m + a) * b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 8e+109)
		tmp = fma(fma(Float64(Float64(pi * angle) * a), -0.011111111111111112, 0.0), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(Float64(b_m + a) * b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 8e+109], N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112 + 0.0), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(N[(b$95$m + a), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.99999999999999985e109

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 59.2

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, \frac{-1}{90}, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, 0\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 7.99999999999999985e109 < angle

   1. Initial program 29.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 27.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot b\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 24.5%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot b\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 54.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;angle \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(angle \cdot \pi\right), a, \left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b\_m + a\right) \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= angle 8e+109)
   (fma
    (* (* -0.011111111111111112 a) (* angle PI))
    a
    (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112))
   (* (* 0.011111111111111112 angle) (* PI (* (+ b_m a) b_m)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if (angle <= 8e+109) {
		tmp = fma(((-0.011111111111111112 * a) * (angle * ((double) M_PI))), a, (((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112));
	} else {
		tmp = (0.011111111111111112 * angle) * (((double) M_PI) * ((b_m + a) * b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (angle <= 8e+109)
		tmp = fma(Float64(Float64(-0.011111111111111112 * a) * Float64(angle * pi)), a, Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(Float64(b_m + a) * b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[angle, 8e+109], N[(N[(N[(-0.011111111111111112 * a), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(N[(b$95$m + a), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.99999999999999985e109

   1. Initial program 58.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 59.2

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto a \cdot \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) + \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right)\right) \cdot a + \frac{1}{90} \cdot \left(\color{blue}{angle} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b + -1 \cdot b\right)\right)\right), a, \frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. Applied rewrites 61.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot a, -0.011111111111111112, \left(\left(\left(0 \cdot b\right) \cdot \pi\right) \cdot angle\right) \cdot 0.011111111111111112\right), \color{blue}{a}, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{90} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{90} \cdot a\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90}\right) \]

      5. lift-PI.f64 61.5

         \[\leadsto \mathsf{fma}\left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(angle \cdot \pi\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   10. Applied rewrites 61.5%

       \[\leadsto \mathsf{fma}\left(\left(-0.011111111111111112 \cdot a\right) \cdot \left(angle \cdot \pi\right), a, \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112\right) \]

   if 7.99999999999999985e109 < angle

   1. Initial program 29.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 27.0

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot b\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 24.5%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot b\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 53.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* (* 0.011111111111111112 angle) PI) (* (- b_m a) (+ a b_m))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return ((0.011111111111111112 * angle) * ((double) M_PI)) * ((b_m - a) * (a + b_m));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return ((0.011111111111111112 * angle) * Math.PI) * ((b_m - a) * (a + b_m));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return ((0.011111111111111112 * angle) * math.pi) * ((b_m - a) * (a + b_m))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(Float64(0.011111111111111112 * angle) * pi) * Float64(Float64(b_m - a) * Float64(a + b_m)))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = ((0.011111111111111112 * angle) * pi) * ((b_m - a) * (a + b_m));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   8. difference-of-squares N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

   11. lower--.f64 53.8

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

4. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-*l* 53.8

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right)} \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \]

   2. pow2 53.8

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \]

   3. pow2 53.8

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \]

   4. difference-of-squares-rev 53.8

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \]

   5. *-commutative 53.8

      \[\leadsto \left(\color{blue}{0.011111111111111112} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\pi} \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   11. lift-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

   12. lift--.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

6. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)} \]
7. Add Preprocessing

Alternative 18: 53.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq 5 \cdot 10^{-200}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) 5e-200)
   (* (* -0.011111111111111112 (* a a)) (* PI angle))
   (* (* 0.011111111111111112 angle) (* PI (* b_m b_m)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= 5e-200) {
		tmp = (-0.011111111111111112 * (a * a)) * (((double) M_PI) * angle);
	} else {
		tmp = (0.011111111111111112 * angle) * (((double) M_PI) * (b_m * b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= 5e-200) {
		tmp = (-0.011111111111111112 * (a * a)) * (Math.PI * angle);
	} else {
		tmp = (0.011111111111111112 * angle) * (Math.PI * (b_m * b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= 5e-200:
		tmp = (-0.011111111111111112 * (a * a)) * (math.pi * angle)
	else:
		tmp = (0.011111111111111112 * angle) * (math.pi * (b_m * b_m))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= 5e-200)
		tmp = Float64(Float64(-0.011111111111111112 * Float64(a * a)) * Float64(pi * angle));
	else
		tmp = Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(b_m * b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= 5e-200)
		tmp = (-0.011111111111111112 * (a * a)) * (pi * angle);
	else
		tmp = (0.011111111111111112 * angle) * (pi * (b_m * b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-200], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision], N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 4.99999999999999991e-200

   1. Initial program 59.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 54.7

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 54.5

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 54.5%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]

   if 4.99999999999999991e-200 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 47.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 52.7

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot {b}^{\color{blue}{2}}\right) \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

      2. lower-*.f64 50.6

         \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot \color{blue}{b}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 53.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot \left({b\_m}^{2} - {a}^{2}\right) \leq 5 \cdot 10^{-200}:\\ \;\;\;\;\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot \left(b\_m \cdot b\_m\right)\right) \cdot angle\right) \cdot 0.011111111111111112\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (if (<= (* 2.0 (- (pow b_m 2.0) (pow a 2.0))) 5e-200)
   (* (* -0.011111111111111112 (* a a)) (* PI angle))
   (* (* (* PI (* b_m b_m)) angle) 0.011111111111111112)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (pow(b_m, 2.0) - pow(a, 2.0))) <= 5e-200) {
		tmp = (-0.011111111111111112 * (a * a)) * (((double) M_PI) * angle);
	} else {
		tmp = ((((double) M_PI) * (b_m * b_m)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(b_m, 2.0) - Math.pow(a, 2.0))) <= 5e-200) {
		tmp = (-0.011111111111111112 * (a * a)) * (Math.PI * angle);
	} else {
		tmp = ((Math.PI * (b_m * b_m)) * angle) * 0.011111111111111112;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	tmp = 0
	if (2.0 * (math.pow(b_m, 2.0) - math.pow(a, 2.0))) <= 5e-200:
		tmp = (-0.011111111111111112 * (a * a)) * (math.pi * angle)
	else:
		tmp = ((math.pi * (b_m * b_m)) * angle) * 0.011111111111111112
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b_m ^ 2.0) - (a ^ 2.0))) <= 5e-200)
		tmp = Float64(Float64(-0.011111111111111112 * Float64(a * a)) * Float64(pi * angle));
	else
		tmp = Float64(Float64(Float64(pi * Float64(b_m * b_m)) * angle) * 0.011111111111111112);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle)
	tmp = 0.0;
	if ((2.0 * ((b_m ^ 2.0) - (a ^ 2.0))) <= 5e-200)
		tmp = (-0.011111111111111112 * (a * a)) * (pi * angle);
	else
		tmp = ((pi * (b_m * b_m)) * angle) * 0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-200], N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 4.99999999999999991e-200

   1. Initial program 59.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 54.7

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      4. pow2 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

      8. lift-PI.f64 54.5

         \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

   7. Applied rewrites 54.5%

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]

   if 4.99999999999999991e-200 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 47.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

      11. lower--.f64 52.7

          \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

   4. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{90} \]

      3. *-commutative N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(\left(\pi \cdot {b}^{2}\right) \cdot angle\right) \cdot \frac{1}{90} \]

      8. pow2 N/A

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \frac{1}{90} \]

      9. lower-*.f64 50.5

         \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot 0.011111111111111112 \]

   7. Applied rewrites 50.5%

      \[\leadsto \left(\left(\pi \cdot \left(b \cdot b\right)\right) \cdot angle\right) \cdot \color{blue}{0.011111111111111112} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 52.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b\_m - a\right) \cdot \left(a + b\_m\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* (* angle PI) 0.011111111111111112) (* (- b_m a) (+ a b_m))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return ((angle * ((double) M_PI)) * 0.011111111111111112) * ((b_m - a) * (a + b_m));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return ((angle * Math.PI) * 0.011111111111111112) * ((b_m - a) * (a + b_m));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return ((angle * math.pi) * 0.011111111111111112) * ((b_m - a) * (a + b_m))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(Float64(angle * pi) * 0.011111111111111112) * Float64(Float64(b_m - a) * Float64(a + b_m)))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = ((angle * pi) * 0.011111111111111112) * ((b_m - a) * (a + b_m));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(a + b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \left(\left(2 \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\left(2 \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. lift-sin.f64 N/A

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   11. lift-cos.f64 N/A

       \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right)} \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   14. lift-/.f64 N/A

       \[\leadsto \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{angle}{180}}\right) \]

3. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot 2\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]

4. Taylor expanded in b around 0

   \[\leadsto \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto b \cdot \left(2 \cdot \left(b \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + 2 \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a + -1 \cdot a\right)\right)\right)\right) + \color{blue}{-2 \cdot \left({a}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot \mathsf{fma}\left(b, \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right) \cdot \left(0 \cdot a\right)\right), b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)} \]

7. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

   2. difference-of-squares-rev N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

   3. pow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

   4. pow2 N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

   8. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{90} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

   9. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(b + a\right)} \cdot \left(b - a\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)} \]

9. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot \left(\left(b - a\right) \cdot \left(a + b\right)\right)} \]
10. Add Preprocessing

Alternative 21: 52.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* 0.011111111111111112 angle) (* PI (* (+ b_m a) (- b_m a)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return (0.011111111111111112 * angle) * (((double) M_PI) * ((b_m + a) * (b_m - a)));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return (0.011111111111111112 * angle) * (Math.PI * ((b_m + a) * (b_m - a)));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return (0.011111111111111112 * angle) * (math.pi * ((b_m + a) * (b_m - a)))

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(0.011111111111111112 * angle) * Float64(pi * Float64(Float64(b_m + a) * Float64(b_m - a))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = (0.011111111111111112 * angle) * (pi * ((b_m + a) * (b_m - a)));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(0.011111111111111112 * angle), $MachinePrecision] * N[(Pi * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   8. difference-of-squares N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

   11. lower--.f64 53.8

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

4. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]
5. Add Preprocessing

Alternative 22: 34.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle)
 :precision binary64
 (* (* -0.011111111111111112 (* a a)) (* PI angle)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle) {
	return (-0.011111111111111112 * (a * a)) * (((double) M_PI) * angle);
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle) {
	return (-0.011111111111111112 * (a * a)) * (Math.PI * angle);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle):
	return (-0.011111111111111112 * (a * a)) * (math.pi * angle)

Julia

b_m = abs(b)
function code(a, b_m, angle)
	return Float64(Float64(-0.011111111111111112 * Float64(a * a)) * Float64(pi * angle))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle)
	tmp = (-0.011111111111111112 * (a * a)) * (pi * angle);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_] := N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} - {a}^{2}\right)}\right) \]

   5. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{{b}^{2}} - {a}^{2}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right) \]

   8. difference-of-squares N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(\color{blue}{b} - a\right)\right)\right) \]

   11. lower--.f64 53.8

       \[\leadsto \left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - \color{blue}{a}\right)\right)\right) \]

4. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot angle\right) \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   4. pow2 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot angle\right) \]

   8. lift-PI.f64 34.7

      \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot angle\right) \]

7. Applied rewrites 34.7%

   \[\leadsto \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(\pi \cdot angle\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))